Number System for Class 9

To understand the topic Number system for class 9, we need to know all about number system including:

  1. What is Number system?
  2. What are the types of numbers?
  3. Example questions to understand these number types?
  4. Practice questions to understand the number system.

Introduction to number system class 9

The collection of numbers is called Number system

Natural Numbers

N

1, 2, 3, 4, 5, ……

Whole Numbers

W

0,1, 2, 3, 4, 5….

Integers

Z

…., -3, -2, -1, 0, 1, 2, 3, …

Rational Numbers

Q

p/q form, where p and q are integers and q is not zero.

Irrational Numbers

Which can’t be represented as rational numbers

Note: every natural number is an integer and 0 is a whole number which is not a whole number.

How to find a rational number between two given numbers a and b?

  • To find a rational number between a and b, find (a+b)/2.

Example: If we want to find a rational number between 3 and 4, the answer is (3+4)/2 = 7/2.

Irrational Numbers

A number is called an irrational number if it can’t be represented in a p/q form, where p and q are integers.

Real Numbers

The collection of all rational and irrational numbers is called real numbers. Real numbers are denoted by R.

Every real number is a unique point on the number line and also every point on the number line represents a unique real number.

Difference between Terminating and Recurring Decimals.

Terminating Decimals

Repeating Decimals

If the decimal expression of a/b terminates. I.e comes to an end, then the decimal so obtained is called Terminating decimals.

A decimal in which a digit or a set of digits repeats repeatedly periodically, is called a repeating decimal.

Example: ¼ =0.25

Example: ⅔ = 0.666… = 0.\(\bar{6}\)

Some Special Characteristics of Rational Numbers:

  • Every Rational number is expressible either as a terminating decimal or as a repeating decimal.
  • Every Terminating decimal is a rational number.
  • Every repeating decimal is a rational number.

Irrational Numbers

  • The non terminating, non repeating decimals are irrational numbers.

Example: 0.0100100001001…

  • Similarly if m is a positive number which is not a perfect square, then \(\sqrt{m}\) is irrational.

Example: \(\sqrt{3}\)

  • If m is a positive integer which i snot a perfect cube, then \(\sqrt[3]{m}\) is irrational.

Example: \(\sqrt[3]{2}\)

Properties of Irrational Numbers:

  • These satisfy the commutative, associative and distributive laws for addition and multiplication.
  • Sum of two irrationals need not be irrational.

Example: (2 + \(\sqrt{3}\)) + (4 – \(\sqrt{3}\)) = 6

  • Difference of two irrationals need not be irrational.

Example: (5 + \(\sqrt{2}\)) – (3 + \(\sqrt{2}\)) = 2

  • Product of two irrationals need not be irrational.

Example: \(\sqrt{3}\) x \(\sqrt{3}\) = 3

  • Quotient of two irrationals need not be irrational.

2\(\sqrt{3}\)/\(\sqrt{3}\) = 2

  • Sum of rational and irrational is irrational.
  • Difference of rational and irrational is irrational.
  • Product of rational and irrational is irrational.
  • Quotient of rational and irrational is irrational.

Real Numbers

A number whose square is non-negative, is called a real number.

  • Real numbers follows Closure property, associative law, commutative law, existence of additive identity, existence of additive inverse for Addition.
  • Real numbers follows Closure property, associative law, commutative law, existence of multiplicative identity, existence of multiplicative inverse, Distributive laws of multiplication over Addition for Multiplication.

Rationalisation

If we have an irrational number, then the process of converting the denominator to a rational number by multiplying the numerator and denominator by a suitable number, is called rationalisation.

Example:

3/\(\sqrt{2}\) = 3/\(\sqrt{2}\) x \(\sqrt{2}\)/ \(\sqrt{2}\) = 3\(\sqrt{2}\)/2

Laws of Radicals:

Let a>0 be a real number, and let p and q be rational numbers, then we have:

i) (ap x aq) = a(p+q)

ii) (ap)q = apq

iii)ap /aq = a(p-q)

iv) ap x bp = (ab)p

Example: Simplify (36)½

Solution: (62)½ = 6(2 x ½) = 61 = 6

Class 9 Number system Extra questions:

Q1) The simplest form of 1.\(\tilde{6}\) is?

Q2) An irrational number between \(\sqrt{2}\) and \(\sqrt{3}\) is?

Q3) Give an example of two irrational numbers whose sum as well as the product is rational.

To understand the mathematical topics of Class 9 more clearly Visit BYJU’S.

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